Group Theory Q: Deriving (10.80) Detail Calc

[cosmology]

See the attachment(from Quantum Mechanics-Symmetries 2nd W.Greiner B.Muller1994W-p345),
how to derive (10.80)
please show me the detail calculation

 

[malawi_glenn]

10.80 is not derived, I think it is an ansatz

 

[denian]

In order to derive equation (10.80) from the attached content, we must first understand the context and notation being used.

Equation (10.80) is part of the mathematical framework of group theory, which is used to study symmetries in quantum mechanics. In particular, it is related to the representation theory of groups, which deals with how the elements of a group can be expressed as matrices acting on a vector space.

In this case, equation (10.80) is specifically related to the Clebsch-Gordan coefficients, which are used to describe the coupling of angular momenta in a quantum system. These coefficients are expressed as matrix elements of operators belonging to the group SU(2), the special unitary group of degree 2.

To derive equation (10.80), we first start with the definition of the Clebsch-Gordan coefficients, which is given by:

C(j1m1,j2m2;JM) = <j1m1,j2m2|JM>

Where j1 and j2 are the angular momenta of two particles, m1 and m2 are their respective quantum numbers, and JM is the total angular momentum of the coupled system.

Next, we use the Wigner-Eckart theorem, which states that the matrix elements of a tensor operator in a given basis can be written as a product of a Clebsch-Gordan coefficient and a reduced matrix element. In this case, the tensor operator is the angular momentum operator J, and the basis is the coupled basis of the two particles.

Using this theorem, we can rewrite equation (10.80) as:

<J1J2JM|J1J2JM> = C(J1J2JM,JM;JM) * <J1J2JM|J1J2JM>

Now, we use the definition of the Clebsch-Gordan coefficients to write:

C(J1J2JM,JM;JM) = <J1J2JM|J1J2JM>

And substituting this into the previous equation, we get:

<J1J2JM|J1J2JM> = <J1J2JM|J1J2JM> * <J1J2JM|J1J2JM>

Which simplifies to:

<J1J2JM|J1J2JM> = <J1J2JM|J1J2JM>

This is the desired result